Infinite Dimensional Analysis, Quantum Probability and Related Topics

We continue our program of coding the whole information of a probability measure into a set of commutation relations canonically associated to it by presenting some characterization theorems for the symmetry and factorizability of a probability measure on ℝ^d in terms of the canonically associated interacting creation, annihilation and number operators.

The Belavkin equation, describing the continuous measurement of the momentum of a quantum particle, is studied. The existence and uniqueness of its solution is proved via analytic tools. A stochastic characteristics method is applied. A rigorous representation of the solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the phase space is also given.

In its most general formulation a quantum kinematical system is described by a Heisenberg group; the "configuration space" in this case corresponds to a maximal isotropic subgroup. We study irreducible models for Heisenberg groups based on compact maximal isotropic subgroups. It is shown that if the Heisenberg group is 2-regular, but the subgroup is not, the "vacuum sector" of the irreducible representation exhibits a fermionic structure. This will be the case, for instance, in a quantum mechanical model based on the 2-adic numbers with a suitably chosen isotropic subgroup.

The formulation in terms of Heisenberg groups allows a uniform treatment of p-adic quantum systems for all primes p, and includes the possibility of treating adelic systems.

A one-parameter symplectic group {e^tÂ}_t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator U_t implementing such a proper canonical transformation gives a projective unitary representation of {e^tÂ}_t∈ℝ on the Boson–Fock space and that U_t can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent with a real-valued function τ_Â(·) such that .

In this paper we continue our studies, initiated in Refs. 2–4, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici–Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Lévy–Khintchine representations (classical and free, respectively).

We study the fluctuation limits of a class of superprocesses with dependent spatial motion on the real line, which give rise to some new Ornstein–Uhlenbeck processes with values of Schwartz distributions.

We revisit the problem of mutually unbiased measurements in the context of estimating the unknown state of a d-level quantum system, first studied (in the framework of quantum information) by Wootters and fields⁷ in 1989 and later investigated by Bandyopadhyay et al.³ in 2001 and Pittenger and Rubin⁶ in 2003. Our approach is based directly on the Weyl operators in the L²-space over a finite field when d=p^r is the power of a prime. When d is not a prime power we sacrifice a bit of optimality and construct a recovery operator for reconstructing the unknown state from the probabilities of elementary events in different measurements.

We identify the representation of the square of white noise obtained by Accardi, Franz and Skeide in [Comm. Math. Phys.228 (2002) 123–150] with the Jacobi field of a Lévy process of Meixner's type.

In this paper we establish a theory of stochastic integration with respect to the basic field operator processes in the Boolean case. This leads to a Boolean version of quantum Itô's product formula and has applications to the theory of dilations of quantum dynamical semigroups.